High School: Functions

Trigonometric Functions HSF-TF.C.9

9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

To add or not to add; that is the question. For some students, it could mean the difference between life and death. We may or may not be upping the drama, but what is Hamlet if not dramatic?

Students should be able to understand written proofs of the addition and subtraction formulas for sine, cosine, and tangent. Upon studying these proofs, they should also be able to reproduce them.

sin(α + β) = sinα cosβ + cosα sinβ

cos(α + β) = cosα cosβ – sinα sinβ

To find the sine, cosine, or tangent of α – β instead of α + β, just switch all the signs on the other side of the equation.

Working through the proofs a few times with the students is usually a good choice. It may be helpful to first review the negative angle identities and cofunctions before working through the proofs. Hopefully, students are well versed in radians by now, but if they aren't, it might be better to work the first few problems in degrees.

Although the goal is for students to understand these proofs, some students might have no choice but to memorize them. Even still, it's easier than memorizing and understanding a Shakespeare monologue, right?

It helps to explain to students that these formulas are very handy if we have angles that are combinations of special angles (30°, 45°, 60°, and their other-quadrant equivalents). For example the exact value of sin(15°) could be found by using sin(45° – 30°). Exact values might not look pretty, but students can't always count on a calculator to be by their side.

Sum and difference identities can come in handy other ways, too. We could derive the cofunction identities and the double angle identities by using the sum and difference identities. Or we can use them to simplify trigonometric expressions like cos(θ + 3π).

They're helpful in solving other proofs as well. And your students will definitely need them when calculus rolls around. Let's just hope they don't descend into a Hamlet-esque madness by then.